176 F. Bethuel - P. Caldiroli - M. Guida
Then, questions Q and Q
1
can be formulated in terms of ordinary differential equations. More precisely, the fact that C has curvature κus at every point us
belonging to C reads 1
¨u = iκu ˙u on I, where i denotes the rotation by
π 2
. Note that the sign of the term of the r.h.s. depends on a choice of orientation, and the curvature might therefore take negative values.
The constraint | ˙u| = 1 might raise difficulties in order to find solutions to Q
and Q
1
. It implies in particular that |I | = length of C, and this quantity is not know
a priori. This difficulty can be removed if we consider instead of 1 the following equivalent formulation
2 |I |
1 2
R
I
| ˙u|
2
ds
1 2
¨u = iκu ˙u on I. To see that 2 is an equivalent formulation of 1, note first that any solution u to 2
verifies 1
2 d
ds | ˙u|
2
= ¨u · ˙u = R
I
| ˙u|
2
ds
1 2
|I |
1 2
κ ui
˙u · ˙u = 0, so that
| ˙u| = C = const. and, introducing the new parametrization vs = usC
, we see that
| ˙v| = 1, and v solves 1. Hence, an important advantage of formulation 2 is that we do not have to impose
any auxiliary condition on the parametrization since equation 2 is independent of the interval I . Thus, we may choose I
= [0, 1] and 2 reduces to 3
¨u = i Luκu ˙u on [0, 1] , where
Lu : =
Z
I
| ˙u|
2
ds
1 2
. Each of the questions Q
and Q
1
has then to be supplemented with appropriate boundary conditions:
u0 = u1, ˙u0 = ˙u1 for Q
or alternatively, to consider RZ instead of [0, 1], and 4
u0 = a, u1 = b, for Q
1
.
1.2. The case of constant curvature
We begin the discussion of these two questions with the simplest case, namely when the function κ is a constant κ
0. It is then easily seen that the only solutions to
Parametric surfaces 177
equations 1 or 3 are portions of circles of radius R =
1 κ
. Therefore, for Q we
obtain the simple answer: the solutions are circles of radius 1κ .
For question Q
1
a short discussion is necessary: we have to compare the distance l
: = |a − b| with the diameter D
= 2R . Three different possibilities may occur:
i l D
, i.e.,
1 2
l κ
1. In this case there is no circle of diameter D containing
simultaneously a and b, and therefore problem Q
1
has no solution. ii l
= D , i.e.,
1 2
l κ
= 1. There is exactly one circle of diameter D containing
simultaneously a and b. Therefore Q
1
has exactly two solutions, each of the half-circles joining a to b.
iii l D
, i.e.,
1 2
l κ
1. There are exactly two circles of diameter D containing
simultaneously a and b. These circles are actually symmetric with respect to the axis ab. Therefore Q
1
has exactly four solutions: two small solutions, symmetric with respect to the axis ab, which are arcs of circles of angle strictly
smaller than π , and two large solutions, symmetric with respect to the axis ab, which are arcs of circles of angle strictly larger than π . Notice that the length
of the small solutions is 2 arccos
1 2
l κ
κ
−1
, whereas the length of the large solutions is 2π
− arccos
1 2
l κ
κ
−1
, so that the sum is the length of the circle of radius R
. As the above discussion shows, the problem can be settled using very elementary
arguments of geometric nature. We end this subsection with a few remarks concerning the parametric formulation,
and its analytical background: these remarks will be useful when we will turn to the general case.
Firstly, we observe that equation 3 in the case κ ≡ κ
is variational: its solutions are critical points of the functional
F
κ
v = Lv − κ
Sv where Lv has been defined above and
Sv : =
1 2
Z
1
i v · ˙v ds .
The functional space for Q is the Hilbert space
H
per
: = {v ∈ H
1
[0, 1], R
2
| v0 = v1} , whereas the functional space for Q
1
is the affine space H
a,b
: = {v ∈ H
1
[0, 1], R
2
| v0 = a, v1 = b} . The functional Sv have a nice geometric interpretation. Indeed, for v belonging to
the space H
per
, Sv represents the signed area of the inner domain bounded by the
178 F. Bethuel - P. Caldiroli - M. Guida
curve Cv = v[0, 1]. Whereas, for v in H
per
or H
a,b
, the quantity Lv is less or equal to the length of Cv and equality holds if and only
| ˙v| is constant. In particular, for v in H
per
, we have the inequality 4π
|Sv| ≤ L
2
v, which is the analytical form of the isoperimetric inequality in dimension two. There-
fore solutions of Q , with κ
≡ κ are also solutions to the isoperimetric problem
sup {Sv | v ∈ H
per
, Lv
= 2πκ
−1
} . This, of course, is a well known fact.
Finally, we notice that the small solutions to Q , in case iii are local min-
imizers of F. More precisely, it can be proved that they minimize F on the set {v ∈ H
a,b
| kvk
∞
≤ κ
−1
} in this definition, the origin is taken as the middle point of ab. In this context, the large solution can then also be analyzed and obtained
variationally, as a mountain pass solution. We will not go into details, since the argu- ments will be developed in the frame of H -surfaces here however they are somewhat
simpler, since we have less troubles with the Palais-Smale condition.
1.3. The general case of variable curvature
